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HelmutMoritz said in e-mail: "... Quantum Theorywas founded independently in two completely differentmathematicalforms by two equally great physicists, Heisenbergand Schroedinger. The miracle was thatboth mathematical formulations were soon recognized to be equivalent:the underlying Hilbert space may be formulated in two equivalentways: as a space of functions or a space if infinite-dimensionalvectors. A periodic function is equivalent to the infinite vectorformed by its Fourier coefficients.

If there had been a collaboration afterWorld War II among

Schroedinger

Heisenberg

Bohm

they might have formulated something similar to

the D4-D5-E6-E7-E8 VoDou Physicsmodel:


The unified theory of

Schroedinger

generalized Einstein's relativity by using4-dimensional geometry with antisymmetric components andconnections.

 

Feynman, in The Feynman Lectures on Gravitation (Addison-Wesley1995), says (on pages 24, 30-32): "... what is the spin of thegraviton?

If the spin were 1/2, or half integral, ... there could be nointerference between the amplitudes of a single exchange, and noexchange ... a half unit of angular momentum cannot be emitted by anobject that remains in the same internal state as it started in ...Thus the spin of the graviton must be integral ...

The rejection of spin-zero theories of gravitation is made on thebasis of the gravitational behavior of binding energies. ... theinteraction energy ... corresponding to the spin-0 field, would beproportional to sqrt( 1 - v^2/c^2 ). In other words, the spin-zerotheory would predict that attraction between masses of hot gas wouldbe smaller than for cool gas. ... the experimental evidence ongravity suggests that the force is greater if the gases are hotter...

A spin-1 theory would be essentially the same as electrodynamics.... one consequence of the spin 1 is that likes repel, and unlikesattract. This is ... a property of all odd-spin theories; ... evenspins lead to attractive forces ...

the spin-2 theory leads to an interaction energy...[between]... two masses of gas ... which has sqrt( 1 -v^2/c^2 ) in the denominator, in agreement with ... the experimentalevidence on gravity ... that the force is greater if the gases arehotter ...

In the theories of scalar, vector,and tensor fields (another wayof denoting spins 0, 1, and 2) the fields are described by scalar,vector, or tensor potential functions:

... assuming that the tensor is anti-symmetric ... would not leadto something resembling gravity, but rather something resemblingelectromagnetism; the six independent components of the antisymmetrictensor would appear as two space vectors. ...".

In Feynman's picture, if the fields are taken to be 4-dimensionalcovariant derivatives of the potentials, then there are 4 times asmany fields as there are potentials, so that for spin 2 thereare:

so that there are 16 general huv spin 2 potentials, 10 of whichare symmetric and give Einstein gravity and 6 of which areantisymmetric and sort of like electromagnetism.

However, I think that Feynman's intuition is wrong about the 6being like "two space vectors". From the point of view that 4x4 realantisymmetric matrices correspond to Spin(1,3) Lorentztransformations of 4-dim Minkowski space, which are the 6 bivectorsof the 1 4 6 4 1 graded Clifford algebra structure

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

it appears that the 6 is like 3 bivector rotations of 1 3 3 1which correspond to spatial rotations in 4-dim spacetime and tospatial angular momentum, plus 3 spatial vectors of 1 3 3 1 whichcorrespond to boosts involving time in 4-dim spacetime and to spatiallinear momentum, giving 4 corresponding to 3 space and 1 time(possibly special conformal) plus 2 others (possibly corresponding todilatation and a phase).

A theory of Schroedinger is described in a29 January 1947 AP newspaper article in part as follows:

"... DUBLIN, Eire, Jan 29 (AP) - ... Dr. Schroedinger said ... "Weknow that a rotating mass, such as the earth, has a magnetic field- now this theory should show why it has. ... The physicist saidthe Einstein theory was "purely geometrical" and that nowelectromagnetism must be included. ... Dr. Schroedinger said hehad arrived at his theory by relying on a "general nonsymmetricaffinity," whereas other scientists had failed to solve theproblem because they used a "symmetrical affinity with only fortycomponent parts instead of a general one with sixty-four[component parts]." ...".


The Feynman-Schroedinger picture and the Cl(8)Clifford algebra picture of the D4-D5-E6-E7-E8VoDou physics model then correspond this way:
Feynman-Schroedinger Clifford Cl(8)  10 symmetric potentials 10 deSitter generators give the 4x10 = 40 components by MacDowell-Mansouri mechanism of Einstein's tensor gravity give Einstein gravity  6 antisymmetric potentials 4 special conformal GraviPhotonsgive Blackett-Sirag 1 scale dilatation graviphoton effect 1 U(1) propagator phase

Considering that Einstein also used the symmetric 40 of the 64h_uv components to model gravity by Einstein's theory ofrelativity,

Did Einstein also use the antisymmetric 24 components to getthe Blackett-Sirag graviphotoneffect ?

I don't have a conclusive answer, but here are some indicationsthat Einstein did NOT get the Blackett-Sirag graviphoton effect, butonly attempted to get ordinary electromagnetism:

1 - upon hearing of Schroedinger's 1947 claim of having used the24 anti-symmetric components to get the Blackett-Sirageffect, "... Einstein ... realised immediately that there wasnothing of merit in Schrödinger's 'new theory' ... Einsteinwrote immediately breaking off the correspondence [withSchroedinger] on unified field theory. ....", according toaSchroedinger web page.

2 - "... The nonsymmetric field extension of Einstein gravity hasa long history. It was originally proposed by Einstein, as a unifiedfield theory of gravity and electromagnetism ... A. Einstein, TheMeaning of Relativity, fifth edition, Princeton University Press,1956. A. Einstein and E. G. Straus, Ann. Math. 47, 731 (1946) ... Butit was soon realized that the antisymmetric part g[uv] in thedecomposition guv = g(uv) + g[uv] could not describephysically the electromagnetic field. ...", acccording to apaper by J. W. Moffat in which he (Moffat) describes his attemptsat using the nonsymmetric field structure to describe, notelectromagnetism, but a generalization of Einstein gravity denoted byNGT (Nonsymmetric Gravitational Theory). Moffat concludes that hisNGT is unsatisfactory, and he abandons it for another (morecomplicated in my opinion) theory.

However, as far as I can see from looking at a few Moffat papers,Moffat never discusses (or even makes reference to) the 1947 theoryof Schroedinger.

In Schroedinger's book Space-Time Structure (Cambridge 1950),Schroedinger says, of the tensor g_ik, that "... hoping that its skewpart (1/2)(g_ik - g_ki) should have something to do withelectromagnetism meets with a certain difficulty. ... The way out ofthis dilemma is ... due to Palatini ... Take the g_ik AND the ...Christoffel brackets ... GAMMA^i_kl as the independent functions, tobe varied ...".

To me Schroedinger's approach looks a lot like Cartan-Einstein,which varies both metric and connection and has torsion. (It isinteresting that Einstein had correspondence with both Cartan andSchroedinger, but apparently insofar as I understand it, neither setof correspondence produced any substantial shared understanding orprogress.)

I think that Schroedinger failed to get ordinary electromagnetism,but may have gotten the Blackett-Sirageffect plus gravity.

If Schroedinger had been content to add in ordinaryelectromagentism by a U(1) gauge field, I think that he would havebeen more successful.

How could Schroedinger have gone beyond the 64-componenttheory to get SU(2) weak force and SU(3) color and U(1)electromagnetism ?


Schroedinger could have used the idea of

Heisenberg

of a 256-component theory.

 

According to Uncertainty the life and science of Werner Heisenberg,by David C. Cassidy (Freeman 1992) at pages 541-545:

"... By 1957, Heisenberg had modified his matter field to form aneight-component type of spinor field ... in February 1957 ... Amathematical ... battle ... broke out between Heisenberg and Pauliover ... the introduction of an indefinite space metric ... later[in 1957] Heisenberg ... stopped in Zurich for a ... visitwith Pauli. Within a few weeks ... Heisenberg ... happened on a verysimple field equation ...[with]... symmetry ... of ... boththe relativistic Lorentz group and the isospin group. Pauli waselated ... The distribution ...[of a preprint]... was set forFebruary 27, 1958. ... Three days before the preprint was to bedistributed, Heisenberg announced the new formula .... at ...Gottingen ... One enthused press agent proclaimed, 'ProfessorHeisenberg and his assistant, W. Pauli, have discovered the basicequation of the cosmos!' ... Pauli had grown increasingly doubtful... until he refused any further support of the theory ... later ...Heisenberg ... wrote 'He criticized many details of my analysis,some, I thought, quite unreasonably.' ... Pauli ... died suddenly ofcancer at the age of 58. ... Heisenberg ... claimed that Platonismhad dominated his thinking throughout his career. ... 'The particlesof modern physics are representations of symmetry groups and to thatextent they resemble the symmetrical bodies ofPlato's philosophy...'. [ It would be natural for Platonic geometryto have been a dominant force in Heisenberg's way of thinking,because Heisenberg was a student of Sommerfeld, and Sommerfeld was astudent of Felix Klein. ]... Heisenberg fell ill ...cancer of the kidneys and gall bladder ... died peacefully at home inMunich on Sunday, February 1, 1976. ...".

In 1959, the Heisenberg paper from which Pauli withdrew wasextended and published by Durr, Heisenberg, Mitter, Schlieder, andYamazaki (Z. Naturforschg. 14a (1959) 441).

Durr and some others have continued to work on it since then, forexample Durr and Saller (Phys. Rev. D22 (1980) 1176) and some otherpublications by Durr, Saller, et. al.

They begin with U(2) = U(1)xSU(2).Then they construct a fundamental field (urfield) Xwhose 4 components are the elements of U(2) = U(1)xSU(2).For example (my examples, not theirs),you might think of them as the identity plus 3 Pauli matrices,or as 4 complex Dirac gammas. Denote the 4 things by:X1   X2   X3   X4Then take the duals of those 4 things:                    X1*  X2*  X3*  X4*These 8 things form their 8-element fundamental urfieldX1   X2   X3   X4   X1*  X2*  X3*  X4*Those 8 things form the following 256 antisymmetric(spinor-like, fermion-like) combinations:
0-grade with 1 element:1 (unity, zero)1-grade with 8 elements:Xi i arbitrary from 1 to 4Xi* i arbitrary from 1 to 42-grade with 6+16+6 = 28 elements:Xi Xj i =/= jXi X*j i,j arbitrary from 1 to 4X*i X*j i =/= j3-grade with 4+24+24+4 = 56 elements:Xi Xj Xk i =/= j =/= kXi Xj X*k i =/= j, k arbitraryXi X*j X*k i arbitrary, j =/= kX*i X*j X*k i =/= j =/= k4-grade with 1+16+36+16+1 = 70 elements:Xi Xj Xk Xm i =/= j =/= k =/= mXi Xj Xk X*m i =/= j =/= k, m arbitraryXi Xj X*k X*m i =/= j, k =/= mXi X*j X*k X*m i arbitrary, j =/= k =/= mX*i X*j X*k X*m i =/= j =/= k =/= m5-grade with 56 elements:...(dual to 3-grade)...6-grade with 28 elements:...(dual to 2-grade)...7-grade with 8 elements:...(dual to 1-grade)...8-grade with 1 element:X1 X2 X3 X4 X*1 X*2 X*3 X*4
The structure has 1+8+28+56+70+56+28+8+1 = 256 complex dimensions,They call it the Proliferated Urfield,andit has the graded structure of the complexified Clifford algebra Cl(8;C). (In this and in what follows, I will sometimes ignore signature complications, such as whether I should use the Clifford algebras Cl(0,8) or Cl(4,4) etc.)

The D4-D5-E6-E7-E8VoDou Physics model is based on the256-real-dimensional Cliffordalgebra Cl(8), using 4 real Diracgammas,

Radical Unification uses 4 complex Dirac gammas.I think that Heisenberg and Durr use complex Dirac gammasbecause complex Dirac gammas were generalizations ofcomplex Pauli matrices, and were found to be useful insimpler, less complete, physics models such as QED.The grade-2 part of real Cl(8) isthe 28-real-dimensional D4 Lie algebra Spin(8),and (prior to dimensional reduction of physical spacetimefrom 8-dim to 4-dim) it is the gauge symmetry groupin the Lagrangian of my physics model.If Radical Unification were only a complexificationof my physics model, it would have a Lagrangianwith 28-complex-dimensional gauge symmetry group Spin(8,C).As Durr says in his paper Radical Unification(which paper is a continuation of the paper by Durr, Heisenberg,Mitter, Schlieder, and Yamazaki in Z. Naturforschg. 14a (1959) 441):"... Symmetry of the urfield LagrangianThe Lagrangian ... has an extremely high symmetry.It is not only invariant under Poincare transformations and dilatations,... but, in fact, under the full 15-parameter conformal group. ...... In addition ... it appears to be invariant under the hugegauge-type group U(1) x SL(4,C) ...".SL(4.C) has 15 complex dimensions,and there is a local Lie algebra isomorphism SL(4,C) = Spin(6,C).

Spin(6,C) is a subgroup of the Spin(8,C) symmetrygroup of the complexified version of the D4-D5-E6-E7-E8VoDou Physics model, so the Lagrangian ofa complexified version of the D4-D5-E6-E7-E8VoDou Physics model has a larger Spin(8,C)symmetry than the symmetry of the Lagrangian of RadicalUnification.

If you were to look at only the real part ofthe complex Radical Unification structure, and its Lagrangian,you would see that the symmetry group would be SL(4,R)which has a local Lie algebra isomorphism SL(4,R) = Spin(3,3).Since 15-real-dimensional Spin(3,3) = SU(2,2) isthe conformal group of non-linear conformal transformationsof 4-real-dimensional Minkowski space,it seems to me that Durr's remark"... The Lagrangian ... is ... invariant under ...... the full 15-parameter conformal group ...".could be interpreted as referring to the symmetry ofthe real part of Radical Unification and its Lagrangian.

Since the compact version of the conformal group,Spin(6) = SU(4), is a 15-real-dimensional subgroup of the28-real-dimensional Spin(8) gauge symmetry group of theD4-D5-E6-E7-E8 VoDouPhysics model, and since I use thenon-compact conformal group Spin(2,4) = SU(2,2) as a component of theU(2,2) = U(1) x SU(2,2) subgroup of Spin(4,4) to (again ignoring somesignature matters) describe gravity by the MacDowell-Mansourimechanism, leaving 28 - 16 - 12 realdimensions to form the standard model SU(3)xSU(2)xU(1).

The Heisenberg-Durr Lagrangian is described by Durr in Radical Unification,in terms of the 8 complex basis elements              X1  X2  X3  X4  X*1 X*2 X*3 X*4as"... an expression which has essentially the structure ofthe 4x4 determinant constructed from X and X* ...... involving also derivatives of the fields(... up to the third derivative) ...".As Durr says,their full Lagrangian appears on its face to have symmetry U(1) x SL(4,C),which is locally isomorphic to U(1) x Spin(6,C),butDurr says that anticommutator structures involving both X and X*cause their Lagrangian symmetry to be reduced to U(1) x SU(2).U(1)xSU(2) is fine for the electroweak force,but is not big enough to include the SU(3) color force. As to the color force problem, Durr says:"... whether soliton-type solutions are possible or not it is,of course, by no means obvious that they will offer a chancefor a dynamical interpretation of the colour property ...".

In the D4-D5-E6-E7-E8VoDou Physics model, you have a naturalSU(3) color force, but the unconfined particles that we see(protonsand pions)are soliton structures made up of confined quarks and gluons, whichis a similar approach to that advocated by Durr.

As to gravity, Durr says:"... There is an extension of the gauge invariance group ...U(1)xSU(2)...which includes the local Lorentz groupif one explicitly introduces the vierbein as independent field.Here then the ... additional "gauge fields" ... connected withthe SL(4,C) invariance of the naive Lagrangian ... arereally "connections" and relate to torsion. ...".

Durr's description is very similar to theMacDowell-Mansourimechanism that I use in theD4-D5-E6-E7-E8 VoDouPhysics model, so we both get gravity withtorsion etc. by effectively gauging "connection" "gaugefields".

3 main points of comparison between Heisenberg-Durr RadicalUnification and the D4-D5-E6-E7-E8 VoDouPhysics model are:

1 -

2 -
3 -
Both approaches get gravity plus torsion, etc., by something like a MacDowell-Mansouri mechanism gauging of "connection" "gauge fields".

If Schroedinger and Heisenberg had gotten together and used thestructure of a real 256-dimensional Cl(8) Clifford algebra, with areal 8-dimensional vector space, then the picturedescribed by Feynman might have been:

The 8 vector components would have been 4 dimensions of physicalspacetime plus 4 dimensions of internal symmetry space

The 28 bivector components would correspond to the 28 dimensionsof Spin(8).

 Also, you might visualize the 16, not as antisymemtricbivectors, but as general 4x4 Feynman-Schroedinger potentials, with10 asymmetric potentials leading to the 40 components of Einsteingravity and 6 antisymmetric potentials including conformalgraviphotons and producing the Blackett-Sirag effect.

 

How could Schroedinger and Heisenberg have quantized sucha theory ?

 


Schroedinger and Heisenberg could have used the idea of

Bohm

of a Quantum Potential.

Bohm's Quantum Potential QuantumTheory is equivalent to Many-Worlds Quantum Theory. To make aconcrete model of Bohm's Quantum Potential, consider:

The space of Possibilities would have the following physicaldegrees of freedom:

Particle-antiparticle pairs of those fermions would haverepresented the 8/\8 = 28 gauge bosons.

The mathematical interpretation of those degrees of freedom wouldhave been similar to that of the D4-D5-E6-E7-E8VoDou Physics model:

Due to Triality, +/- pairs of thosehalf-spinors could represent the vector /\ vector = 8 /\ 8 = 28bivectors.

Therefore, the Quantum Potential should be described by thephysics of World Lines moving in 8+8+8 = 24-dim space.

If the World Lines are regarded as (closed unoriented bosonic)strings, and if 1+1 dimensions are added to the 24-dim space to giveit Minkowski-like structure, then

the Quantum Theory would hve been theString Theory of closed unoriented bosonic strings in 1+25 = 26-dimspace.

 

As noted by Rey,and by Horowitz andSusskind, that theory may have a 27-dimensional M-theory,related to the 27-dimensional Jordanalgebra J3(O) and its 26-dimensional traceless subalgebraJ3(O)o.

 


If Schroedinger,Heisenberg, and Bohmhad collaborated actively immediately after World War II, they mighthave produced a truly unified theory of everything.

Instead, they remained in their separate worlds:

 


Unified Theories and Hilbert, Kaluza, et al

According to Thall'sHistory of Quantum Mechanics: "...David Hilbert ... professor of mathematics at the University ofGottingen ... suggested to Heisenberg that he find the differentialequation that would correspond to his matrix equations. Had hetaken Hilbert's advice, Heisenberg may have discovered theSchrodinger equation before Schrodinger. When mathematicians provedHeisenberg's matrix mechanics and Schrodinger's wave mechanicsequivalent, Hilbert exclaimed, "Physics is obviously far toodifficult to be left to the physicists ..." ...".

According to PeterWoit: "... By 1925, Hilbert was getting old (63)....".

According to Hilbert'sFoundation of Physics: From a Theory of Everything to a Constituentof General Relativity, by Jü\urgen Renn and John Stachel, MaxPlanck Institute for the History of Science, preprint 118 (1999),ISSN 0948-9444: "... In 1924 Hilbert published anotherrevised version ... Hilbert again claims ... that there is anecessary connection between the theories of Mie ... electromagnetism... and Einstein ... gravitation ... In spite of the reassertion of... Hilbert['s] ... programmatic goal of providingfoundations for all of physics, this theory now was, in effect,transformed into a variation on the themes of general relativity. ...The nature of this source term can be expressed on the level ofthe Lagrangian ... but this relation is in no way peculiarMie's theory. ... Mie's original theory is in fact not gaugeinvariant ... the field equations can only hold ... if ... thetheory is gauge invariant, i.e. the potentials themselves do notenter the field equations ... Hilbert ... did not derive the identityfor gauge-invariant electromagnetic Lagrangians ...".

According to Modern Kaluza-Klein Theories, byApplequist, Chodos, and Freund (Addison-Wesley 1987) and papersreprinted therein: "... In 1914 ... Nordstrom ...[ basedon ]... theories developed by ... Mie and ... Nordstrom ...proceeded to unify ... gravitation with Maxwell's theory ...[ byassuming ]... scalar gravity in our four-dimensional world to bea remnant of an abelian gauge theory in a five-dimensional ...space-time. ... Kaluza in 1919 ... proposed that one pass to anEinstein-type theory of gravity in five dimensions, form whichordinary four-dimensional Einstein gravity and Maxwellelectromagnetism are to be obtained upon imposing a cylindricalconstraint. ... [In] 1938 ... Einstein and ... Bergmann...[said]... two fairly simple and natural attempts toconnect gravitation and electricity by a unitary field theory havebeen made, one by Weyl [ gauge theory ], the other by Kaluza[ higher dimensions ]....".

In his paper Extra gaugefield structure uncovered in the Kaluza-Klein framework, Class.Quantum Grav. 3 (1986) L99-L105, N. A. Batakis says: "... In astandard Kaluza-Klein framework, M4 x CP2 allows the classicalunified description of an SU(3) gauge field with gravity. However,the possibility of an additional SU(2) x U(l) gauge field structureis uncovered. ... As a result, M4 x CP2 could conceivablyaccommodate the classical limit of a fully unified theory for thefundamental interactions and matter fields. ...".

 


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